A Comprehensive Overview on the Formation of Homomorphic Copies in Coset Graphs for the Modular Group

Abstract

This work deals with the well-known group-theoretic graphs called coset graphs for the modular group G and its applications. The group action of G on real quadratic fields forms infinite coset graphs. These graphs are made up of closed paths. When M acts on the finite field Zp, the coset graph appears through the contraction of the vertices of these infinite graphs. Thus, finite coset graphs are composed of homomorphic copies of closed paths in infinite coset graphs. In this work, we have presented a comprehensive overview of the formation of homomorphic copies.

Number of Distinct Fragments in Coset Diagrams for P S L 2 , Z

“…(k (11,l 3 ) ,l 3 ) has |D (k (11,l 3 ) ,l 3 ) . Now, we prove that (i) for α l 3 + β 1 � 1(mod 2), all the elements in F l 3 11 are distinct and no one is the mirror image of itself and (ii) for α l 3 + β 1 � 0(mod 2), all the elements in F l 3 11 are distinct and Γ 11 ((α l 3 +β 1 )/2,l 3 ) is the only fragment which is orientally the same as its mirror image. For this, let Γ 11 (m,l 3 ) and Γ 11…”

“…So, there are total S pairs of joining vertices for Ω and Ω * . In the literature, the question that how many pairs of connecting vertices form the fragment Ω is answered for the pairs of circuits of length-2 [15] and contracting vertices produce the homomorphic image Γ is responded for the circuit of length-4 [11] under certain conditions. We have solved this problem for the pair of circuits of length 6.…”

Revisiting the action of a subgroup of the modular group on imaginary quadratic number fields

“…The study of the action of the modular group PSL(2, Z) and its subgroups on various commutative algebraic structures has been common since the works of Graham Higman and his collaborators in the 1980s ( [8], [11], [12]). Moreover, various tracks of applications of the action of the modular group (especially in multimedia security) have been appearing recently, see for instance [1], [14], [16], [17], and [18]. This indeed suggests a promising future for the applications of this seemingly purely mathematical topic.…”

On q -Rung Orthopair Fuzzy Subgroups

“…Theory of groups is one of the prominent branches of mathematics with numerous applications in physics [15], chemistry [16], cryptography [17][18][19], differential equations [20], and graph theory [21,22]. Rosenfeld [23] initiated the study of fuzzy subgroups.…”